NMR sensing methods are based on the interaction between certain kinds of atomic nuclei and an applied magnetic field. Specifically, atomic nuclei such as those of hydrogen atoms having angular momentum ("spin") and thus a magnetic moment tend to align themselves with the applied field the same way a compass needle aligns with the Earth's magnetic field. This alignment process is primarily determined by the parameters of the applied magnetic field, such as its amplitude, frequency and direction. When the applied magnetic field is turned off, the nuclei which were aligned gradually return to their initial state. This process known as relaxation generates a measurable signal which can be picked up and recorded by sensor instruments. The relaxation signal carries useful information about the number of reacting nuclei, the rate of exchange of energy between the nuclei and the surrounding material, the composition and structure of this material, etc.
To maximize the amount of useful information about the underlying structure of the material under investigation, NMR sensing devices typically apply controlled sequences of magnetic field pulses to a given volume of the material. The sensing devices then detect and record the received "echo" signals and use signal processing algorithms to extract the desired information from the recorded measurement signals.
For example, nuclear magnetic logging tools such as those disclosed in U.S. Pat. Nos. 4,710,713; 4,717,876; 4,717,877 and 4,717,878, assigned to the same assignee and incorporated herein by reference, measure the nuclear magnetic relaxation rates of hydrogen atoms in the pore spaces of earth formations. To this end, a pre-determined high-energy NMR pulse sequence is applied by the tool to the earth formation of interest. The pulse sequence transmits energy at given frequencies thereby disturbing the equilibrium of the hydrogen atoms within the formation. Upon termination of the pulse sequence, the logging tool monitors the returning echo signals as the hydrogen atoms return back to an equilibrium state. The received signals are used to estimate critical parameters in oil exploration which may be properties of the earth formation, such as its porosity, permeability, the irreducible water saturation, residual oil saturation, etc.
To determine the properties of the material under investigation, some of the most frequently used parameters in the analysis of the measured NMR echo signals include: the amplitude of the signal at time t=0 immediately after the magnetic field is turned off; the spin-lattice relaxation time constant T.sub.1, also known as longitudinal relaxation time; and the spin-spin relaxation time constant T.sub.2.
The initial amplitude of the response signal is of special importance in NMR sensing because it directly correlates with the number of hydrogen atoms in the formation (and thus to the probability of finding oil within an earth formation). The T.sub.1 constant indicates the time required for the system to come to a complete thermal equilibrium with its surroundings upon termination of the applied magnetic field. Its value is determined by the molecular environment and typically ranges from 10 to 1000 milliseconds for different types of rocks. The spin-spin relaxation time constant T.sub.2 is indicative of the phase degeneration among signals from individual nuclei due to inhomogeneities in the earth's magnetic field throughout the sample, the type and quantity of fluid, its temperature, viscosity and distribution within the formation. Well known formulas can be applied to derive from these parameters the relevant information regarding the structure of the material under investigation. (See A. Timur, "Pulsed Nuclear Magnetic Resonance Studies of Porosity, Movable Fluid and Permeability of Sandstones" Journal of Petroleum Technology, Jun. 1969, pp. 775-786).
A number of problems are associated, however, with the practical determination of these and other parameters of the received NMR echo signals. Such problems may arise from erroneous signal measurements due to tool calibration inaccuracies, time varying magnetic field pulse parameters, non-negligible logging speed, and especially measurement noise. In addition, only the T.sub.2 relaxation time (or rather the related T.sub.2 * time) of the echo signal is a directly measurable quantity. On the other hand, because of the large magnitudes of the magnetic fields which are used in well logging, the sensing electronics has a "dead time" immediately following the pulse sequence, during which time the echo signal is not observable. Thus, the initial amplitude of the received signal cannot be measured directly and has to be determined by the use of mathematical modeling. Other parameters of interest, such as the T.sub.1 constant, are similarly not available for direct observation and need to be derived from the measured signal by the use of mathematical models. It is thus clear that in the practice of NMR sensing it is necessary to process noise corrupted measurement signals, and is important to derive accurate mathematical models of these signals in order to properly analyze them.
Several approaches have been proposed in the past in this respect. A number of prior art techniques reduce systematic measurement errors by optimizing the parameters of the applied magnetic field. A survey of the available techniques, including the "inversion recovery"; "preparation recovery"; "progressive saturation" and "magnetization conserving" techniques is presented in Levy and Peat, "The Experimental Approach to Accurate Carbon-13 Spin-Lattice Relaxation Measurements," Journal of Magnetic Resonance 18, pp. 500-521, 1975. Pulse sequences having optimal measurement parameters, such as the Carr-Purcell-Meiboom-Gill (CPMG) sequence and modifications of it, have also been developed and are currently accepted as industry standards.
Other prior art NMR sensing methods rely on increasing the accuracy of the obtained measurements by increasing the signal-to-noise (SNR) ratio of the obtained signal. For instance, U.S. Pat. No. 4,631,480 to Young, discloses a method to reduce the measurement noise by using a quantity of a material of known NMR properties in the region under investigation to enhance the measured response signal.
In a different approach, improvements have been made in the mathematical modeling of the measurement signal and the use of signal processing algorithms to reduce the effects of the noise. The simplest signal model frequently used in NMR sensing assumes that the process of relaxation of most materials can be represented by a single exponential decay function. However, research has shown that this model is inadequate to describe the actual processes and may lead to erroneous results. Therefore, most advanced NMR sensing techniques use models in which the measurement signal V(t) obtained in NMR sensing is represented as a sum of independent, decaying exponential signals: ##EQU1## where .phi. is a constant, n is the order of the assumed model, and A.sub.i and T.sub.i are the unknown amplitudes and time constants associated with the relaxation process of different materials or pore sizes within the volume under investigation.
While the approximation of NMR echo signals using the model above is intuitively clear and accurately represents the underlying processes of relaxation in an earth formation, its practical application is associated with considerable difficulties. The reason is that in order to obtain good results, one has to first estimate the order n of the approximation, and then solve a set of non-linear equations for the unknown parameters. As well known in the art, non-linear equations are relatively complex to solve and, in addition, their solutions are frequently unstable and sensitive to small changes in the initial conditions or the measurement noise.
Prior art solutions seek a solution to the problem of mathematical modeling the received echo signals by the use of several techniques, including the graphical "backward projection" technique, (See for example Van Liew, "Semilogarithmic Plots of Data Which Reflects a Continuum of Exponential Processes," Science, 138, 1962); the use of non-linear regression analysis of the measurement signal; non-linear least square fit routines, as disclosed in U.S. Pat. No. 5,023,551, and others. Other prior art techniques include a variety of signal modeling techniques, such as polynomial rooting, singular value decomposition (SVD) and miscellaneous refinements thereof, to obtain a better approximation of the received signal. A common problem with prior art signal models is that their derivation is computationally intensive and rarely takes into account user knowledge about the actual measurement process.
Consider for instance U.S. Pat. No. 4,973,111 to Haacke which describes a method for parametric image reconstruction from sampled NMR measurement data. In the proposed technique, the desired object function is approximated by a series of known model functions having a finite number of unknown parameters. Because the direct equations are highly non-linear, the problem is simplified by using all-pole parameter estimation, in which the unknown parameters are the roots of a polynomial equation. The coefficients of this equation are obtained as the solution vector to a system of linear prediction equations which involve the received measurement data. The solution of the linear prediction system, as suggested in Haacke, is found by applying SVD decomposition to the linear prediction data matrix of the measurement signal. This approach is shown to reduce the effects of the measurement noise and estimate the order of the model functions, resulting in an approximation which is guaranteed to be optimal in a linear least squares sense.
Due to the large size of the involved matrices, however, the method is computationally quite intensive and while suitable for off-line processing does not lend itself to real-time applications of NMR well logging. In addition, the method does not take into account information about the material under investigation or the measurement process, which can be used to simplify the computations.
Thus, notwithstanding the advances in the prior art, it is perceived that the problems involved in the parameter model estimation used in NMR sensing methods for well logging have not yet been resolved. In particular, no simple methods have been proposed to take advantage of prior knowledge about the structure of the investigated material and the signal-to-noise (SNR) ratio of the received echo signal. Also, no efficient solutions have been proposed to combine advanced mathematical models with simple signal processing algorithms to increase the accuracy and numerical stability of the parameter estimates. Finally, existing solutions require the use of significant computational power which makes the practical use of those methods inefficient, and frequently impossible to implement in real-time applications.